The Lebesgue measure $\lambda$ is a function on a subset of the power set of real numbers $\mathbb{R}$ that satisfies the following properties (among others):

(i) $\lambda$ is finitely additive: If $A$ and $B$ are disjoint measurable sets then $\lambda(A \cup B) = \lambda(A) + \lambda(B)$;

(ii) $\lambda$ is defined on a sigma algebra and is countably
sub-additive: $\lambda\!\left(\bigcup_{i = 1}^{\infty} A_i\right) \leq \sum_{i = 1}^{\infty} \lambda(A_i)$;

(iii) $\lambda$ is translation invariant: $\lambda(A + c) =
\lambda(A)$ for any constant $c$ where $A + c = \{a+c \mid a \in
A\}$;

(iv) $\lambda$ respects scaling: $\lambda(cA) = c\lambda(A)$
for any constant $c$ where $cA = \{ca \mid a \in A\}$.

Whether there exist non-measurable sets depends on the assumed model of set theory. In the Solovay model--a model of ZF excluding C(hoice)--every subset of $\mathbb{R}$ is measurable. However, in ZF+C there exist sets that are not Lebesgue measurable (cf. Vitali set). I assume ZF+C for now.

Let us say that $g$ extends $f$ over $\mathbb{R}$ provided $g(A) = f(A)$ for all $f$-measurable sets $A$ and $g(B)$ is defined for at least one non-$f$-measurable set $B \subseteq \mathbb{R}$. I am interested in finding maximal functions (with respect to extension) that mostly satisfy the nice properties of the Lebesgue measure. Toward that end ...

Question: Which of the above properties can be dropped so that there exists some function in $ZF+C$ satisfying the other properties and extending the Lebesgue measure $\lambda$ over $\mathbb{R}$?